Problem: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{2k^2 - 98}{k^3 + 9k^2 + 14k}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {2(k^2 - 49)} {k(k^2 + 9k + 14)} $ $ t = \dfrac{2}{k} \cdot \dfrac{k^2 - 49}{k^2 + 9k + 14} $ Next factor the numerator and denominator. $ t = \dfrac{2}{k} \cdot \dfrac{(k + 7)(k - 7)}{(k + 7)(k + 2)}$ Assuming $k \neq -7$ , we can cancel the $k + 7$ $ t = \dfrac{2}{k} \cdot \dfrac{k - 7}{k + 2}$ Therefore: $ t = \dfrac{ 2(k - 7)}{ k(k + 2)}$, $k \neq -7$